Question

1) Consider X a normal random variable with mean 4 and standard deviation 2. Given that P(X<6)=0.841345 , compute P(2<= x <= 6)

2)Consider X a normal random variable with mean 10 and standard deviation 4. Given that P(x>9)=0.598708 and P(x<12)=0.691464 . Compute P(8< x < 11).

Answer 1

1) Consider X a normal random variable with mean 4 and standard deviation 2. Given that P(X<6)=0.841345 , compute P(2<= x <= 6)

P(2≤X≤6) = P(X<6) – P(X<2)

We are given

P(X<6)=0.841345

Now find P(X<2)

Z = (X – mean)/SD

Z = (2 – 4)/2

Z = -1

P(Z<-1) = P(X<2) = 0.158655

(by using z-table or excel)

P(2≤X≤6) = P(X<6) – P(X<2)

P(2≤X≤6) = 0.841345 - 0.158655

P(2≤X≤6) = 0.68269

Required probability = 0.68269

2)Consider X a normal random variable with mean 10 and standard deviation 4. Given that P(x>9)=0.598708 and P(x<12)=0.691464 . Compute P(8< x < 11).

P(8<X<11) = P(X<11) – P(X<8)

Find P(X<11)

Z = (X – mean)/SD

Z = (11 - 10)/4

Z =0.25

P(Z<0.25) = P(X<11) = 0.598706

(by using z-table)

Now find P(X<8)

Z = (X – mean)/SD

Z = (8 - 10)/4

Z =-0.5

P(Z<-0.5) = P(X<8) = 0.308538

(by using z-table)

P(8<X<11) = P(X<11) – P(X<8)

P(8<X<11) = 0.598706 - 0.308538

P(8<X<11) = 0.290168

Required probability = 0.290168